Add relative and absolute tolerance for rank.

stdlib/LinearAlgebra/src/deprecated.jl

@@ -1,3 +1,4 @@
 # This file is a part of Julia. License is MIT: https://julialang.org/license
 
-using Base: @deprecate, depwarn
+# To be deprecated in 2.0
+rank(A::AbstractMatrix, tol::Real) = rank(A,rtol=tol)

stdlib/LinearAlgebra/src/generic.jl

@@ -712,13 +712,15 @@ end
 ###########################################################################################
 
 """
-    rank(A[, tol::Real])
+    rank(A::AbstractMatrix; atol::Real=0, rtol::Real=atol>0 ? 0 : n*ϵ)
+    rank(A::AbstractMatrix, rtol::Real) = rank(A; rtol=rtol)  # to be deprecated in Julia 2.0
 
 Compute the rank of a matrix by counting how many singular
-values of `A` have magnitude greater than `tol*σ₁` where `σ₁` is
-`A`'s largest singular values. By default, the value of `tol` is the smallest
-dimension of `A` multiplied by the [`eps`](@ref)
-of the [`eltype`](@ref) of `A`.
+values of `A` have magnitude greater than `max(atol, rtol*σ₁)` where `σ₁` is
+`A`'s largest singular value. `atol` and `rtol` are the absolute and relative
+tolerances, respectively. The default relative tolerance is `n*ϵ`, where `n`
+is the size of the smallest dimension of `A`, and `ϵ` is the [`eps`](@ref) of
+the element type of `A`.
 
 # Examples
 ```jldoctest
@@ -728,16 +730,21 @@ julia> rank(Matrix(I, 3, 3))
 julia> rank(diagm(0 => [1, 0, 2]))
 2
 
-julia> rank(diagm(0 => [1, 0.001, 2]), 0.1)
+julia> rank(diagm(0 => [1, 0.001, 2]), rtol=0.1)
 2
 
-julia> rank(diagm(0 => [1, 0.001, 2]), 0.00001)
+julia> rank(diagm(0 => [1, 0.001, 2]), rtol=0.00001)
 3
+
+julia> rank(diagm(0 => [1, 0.001, 2]), atol=1.5)
+1
 ```
 """
-function rank(A::AbstractMatrix, tol::Real = min(size(A)...)*eps(real(float(one(eltype(A))))))
+function rank(A::AbstractMatrix; atol::Real = 0.0, rtol::Real = (min(size(A)...)*eps(real(float(one(eltype(A))))))*iszero(atol))
+    isempty(A) && return 0 # 0-dimensional case
     s = svdvals(A)
-    count(x -> x > tol*s[1], s)
+    tol = max(atol, rtol*s[1])
+    count(x -> x > tol, s)
 end
 rank(x::Number) = x == 0 ? 0 : 1
 

stdlib/LinearAlgebra/test/generic.jl

@@ -195,6 +195,9 @@ end
 
 @test rank(fill(0, 0, 0)) == 0
 @test rank([1.0 0.0; 0.0 0.9],0.95) == 1
+@test rank([1.0 0.0; 0.0 0.9],rtol=0.95) == 1
+@test rank([1.0 0.0; 0.0 0.9],atol=0.95) == 1
+@test rank([1.0 0.0; 0.0 0.9],atol=0.95,rtol=0.95)==1
 @test qr(big.([0 1; 0 0])).R == [0 1; 0 0]
 
 @test norm([2.4e-322, 4.4e-323]) ≈ 2.47e-322